As previously found [1]}, [2]}, a resonance between diffractive radiation and the DSW can occur
when the
phase velocity of the diffractive radiation equals the velocity \(s_{+}\)  of the lead solitary wave of the DSW. As can be
seen from Figure REF (b) this resonant radiation is short wave relative to the DSW, so that the appropriate
linear dispersion relation is (REF ). Resonance then occurs for the wavenumber \(k_{r}\) , with
\(k_{r} = s_{+} + \left( s_{+}^{2} - \frac{4}{q}u_{+}^{2} \right)^{1/2} ,\) 
